[SOLVED] Elliptic functions

Consider the function $\displaystyle w(z)=\int_1^z\frac{dt}{t}$ where the integral is along any rectifiable path from $\displaystyle 1$ to $\displaystyle z$. Clearly this is not a single-valued function of $\displaystyle z$, as the value of the integral will depend on the winding number of the path around $\displaystyle 0$. Thus $\displaystyle w(z)$ is defined up to an integral multiple of $\displaystyle 2\pi i$. Now consider the inverse function $\displaystyle z(w)$; we have $\displaystyle z(w+2\pi i)=z$ and therefore $\displaystyle z$ is a periodic function. (It's the exponential!)

Now consider the function $\displaystyle w(z)=\int_1^z\left(\frac{1}{t}+\frac{i}{t-i}\right)dt$. Now $\displaystyle w$ is defined up to $\displaystyle 2\pi i m + 2\pi i n$ where $\displaystyle m,n \in \mathbb{Z}$. Explain why, mimicking the construction of the inverse function as above, we do not obtain an elliptic function.